Optimal. Leaf size=56 \[ \frac {3 \text {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{b c \sqrt [3]{c \cos (a+b x)} \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722}
\begin {gather*} \frac {3 \sin (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right )}{b c \sqrt {\sin ^2(a+b x)} \sqrt [3]{c \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rubi steps
\begin {align*} \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx &=\frac {3 \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{b c \sqrt [3]{c \cos (a+b x)} \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.95 \begin {gather*} \frac {3 \cot (a+b x) \text {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{b (c \cos (a+b x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \cos \left (b x +a \right )\right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \cos {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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